SOLUTION: Write 5/(6-2i) in a+bi form.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Write 5/(6-2i) in a+bi form.      Log On


   

Question 72133This question is from textbook Advanced Algebra
: Write 5/(6-2i) in a+bi form. This question is from textbook Advanced Algebra

Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
5%2F%286-2i%29
%285%2F%286-2i%29%29%28%286%2B2i%29%2F%286%2B2i%29%29Multiply both top and bottom by the complex conjugate of 6-2i
%285%2A%286%2B2i%29%29%2F%2836%2B12i-12i-4%28-1%29%29FOIL the denominator (remember i%5E2=-1)
%285%2A%286%2B2i%29%29%2F%2836%2Bcross%2812i-12i%29-4%28-1%29%29The i terms cancel in the denominator
%285%2A%286%2B2i%29%29%2F%2840%29%29Distribute the 5 among the parenthesis
%2830%2B10i%29%2F%2840%29%29
30%2F40%2B%2810%2F40%29i
3%2F4%2B%281%2F4%29iHere's the answer in a+bi form where a=3/4 and b=1/4

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
5%2F%286-2i%29
.
to get this to the form a + bi begin by multiplying the given term by:
.
%286+%2B+2i%29%2F%286%2B2i%29
.
Note that this is equivalent to multiplying the given term by 1 because %286+%2B+2i%29%2F%286%2B2i%29
equals 1.
.
The numerator multiplication of the 5 times the (6 + 2i) results in 30 + 20i.
.
Then the denominator multiplication of %286+-+2i%29%2A%286+%2B+2i%29 results in:
.
+36+-+12i+%2B+12i+-4%2Ai%5E2
.
The -12i and + 12i cancel each other out. Then recall that i%5E2 by definition is -1.
Substituting -1 for i%5E2 leads to:
.
36+-4%2A%28-1%29+=+36+%2B+4+=+40
.
This is the denominator ... +40. From above the numerator is 30 + 20i. So the answer is:
.
%2830%2B20%2Ai%29%2F40+=+%283%2F4%29+%2B+%281%2F2%29%2Ai
.
The answer to your problem is %283%2F4%29+%2B+%281%2F2%29%2Ai where a+=+%283%2F4%29 and b=+%281%2F2%29
.
Hope this helps you to understand complex numbers. Notice how you can eliminate complex
numbers in the denominator by multiplying the denominator by the same complex number with
a change in signs between the real and imaginary parts. This converts the denominator
to a real number.
.